GUIDE WAVE ANALYSIS AND FORECASTING - WMO

7.1 Introduction
The evolution of waves in deep water, as treated in
Chapter 3, is dominated by wind and by propagation
along straight lines (or great circles on the globe). When
waves approach the coast, they are affected by the
bottom, currents and, very close to shore, also by obstacles,
such as headlands, breakwaters, etc., the effects of
which usually dominate — surpassing the effects of the
local wind — and the resulting wave propagation is no
longer along straight lines.
When approaching the continental shelf from the
ocean the initial effects of the bottom on the waves are
not dramatic. In fact, they will hardly be noticeable until
the waves reach a depth of less than about 100 m (or
rather, when the depth is about one-quarter of the wavelength).
The first effect is that the forward speed of the
waves is reduced. This generally leads to a slight turning
of the wave direction (refraction) and to a shortening of
the wavelength (shoaling) which in turn may lead to a
slight increase or decrease in wave height. Wind generation
may be enhanced somewhat as the ratio of wind
speed over wave speed increases when the waves slow
down. However, this is generally masked by energy loss
due to bottom friction. These effects will be relatively
mild in the intermediate depths of around 100 m but they
will accumulate so that, if nothing else happens, they
will become noticeable as the distances increase.
When the waves approach the coast from intermediate
water depth and enter shallow water of 25 m or less,
bottom effects are generally so strong (refraction and
dissipation) that they dominate any wind generation. The
above effects of refraction and shoaling will intensify and
energy loss due to bottom friction will increase. All this
suggests that the wave height tends to decrease but propagation
effects may focus energy in certain regions,
resulting in higher rather than lower waves. However, the
same propagation effects may also defocus wave energy,
resulting in lower waves. In short, the waves may vary
considerably as they approach the coast.
In the near-shore zone, obstacles in the shape of
headlands, small islands, rocks and reefs and breakwaters
are fairly common. These obviously interrupt the
propagation of waves and sheltered areas are thus
created. The sheltering is not perfect. Waves will penetrate
such areas from the sides. This is due to the
short-crestedness of the waves and also due to refraction
which is generally strong in near-shore regions. When
the sheltering is very effective (e.g. behind breakwaters)
waves will also turn into these sheltered regions by radi-
CHAPTER 7
WAVES IN SHALLOW WATER
L. Holthuijsen: editor
ation from the areas with higher waves (diffraction).
When finally the waves reach the coast, all shallow
water effects intensify further with the waves ending up
in the surf zone or crashing against rocks or reefs.
Very often near the coast the currents become appreciable
(more than 1 m/s, say). These currents may be
generated by tides or by the discharge from rivers entering
the sea. In these cases the currents may affect waves in
roughly the same sense as the bottom (i.e. shoaling,
refraction, diffraction, wave breaking). Indeed, waves
themselves may generate currents and sea-level changes.
This is due to the fact that the loss of energy from the
waves creates a force on the ambient water mass, particularly
in the breaker zone near a beach where long-shore
currents and rip-currents may thus be generated.
7.2 Shoaling
Shoaling is the effect of the bottom on waves propagating
into shallower water without changing direction.
Generally this results in higher waves and is best demonstrated
when the wave crests are parallel with the bottom
contours as described below.
When waves enter shallow water, both the phase
velocity (the velocity of the wave profile) and the group
velocity (the velocity of wave energy propagation) change.
This is obvious from the linear wave theory for a sinusoidal
wave with small amplitude (see also Section 1.2.5):
(7.1)
with wavenumber k = 2π/λ (with λ as wavelength),
frequency ω = 2π/T (with T as wave period), local
depth, h, and gravitational acceleration, g. These
waves are called “dispersive” as their phase speed
depends on the frequency. The propagation speed of
the wave energy (group speed cg) is cg = βcphase with
β = 1 ω g
cphase
= = tanh( kh)
k k
/2 + kh/sinh (2kh). For very shallow water (depth
less than λ/25) both the phase speed and the group speed
reduce to cphase = cg = √(gh), independent of frequency.
These waves are therefore called “non-dispersive”.
The change in wave height due to shoaling (without
refraction) can be readily obtained from an energy balance.
In the absence of wave dissipation, the total transport of
wave energy is not affected, so that the rate of change
along the path of the wave is zero (stationary conditions):
d
ds cg E ( ) = 0,
(7.2)
where c gE is the energy flux per unit crest length (energy
E = ρ wgH 2 /8, for wave height H) and s is the coordinate